# Let be the corresponding real eigenvectors and be the

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Unformatted text preview: eigenvectors and , be the real and imaginary parts of the complex conjugate eigenvectors. The transformation matrix is nonsingular and where . The solution of the initial value problem will involve the matrix exponential . In this way we compute the matrix exponential of any matrix that is diagonalizable. 2.6 Multiple Eigenvalues The commutator of commute. Fact. If and and is . If the commutator is zero then , then and . .□ Proof. Generalized Eigenspaces Let where . Recall that eigenvalue . This can be rewritten as and eigenvector . Suppose has algebraic multiplicity 1. Then the associated eigenspace is satisfy 4 Chapter 2 part B . A space is invariant under the action of invariant under by the fact above. Suppose is an eigenvalue of eigenspace of as if implies . For example, with algebraic multiplicity is . Define the generalized . The symbol refers to generalized eigenspace but coincides with eigenspace if A nonzero solution to is a generalized eigenvector of . Lemma 2.5 (Invariance). Each of the generalized eigenspaces of...
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